Optimal. Leaf size=248 \[ \sqrt {a x-b} \left (\frac {\sqrt {\sqrt {a x-b}+a x}}{a}-\frac {2}{a}\right )-\frac {3 \sqrt {\sqrt {a x-b}+a x}}{2 a}+\frac {(4 b-19) \log \left (2 \sqrt {a x-b}-2 \sqrt {\sqrt {a x-b}+a x}+1\right )}{4 a}+\frac {2 \log \left (2 \sqrt {a x-b} \sqrt {\sqrt {a x-b}+a x}-\sqrt {\sqrt {a x-b}+a x}-2 a x+1\right )}{a}-\frac {4 (2 b-3) \tan ^{-1}\left (\frac {-2 \sqrt {a x-b}+2 \sqrt {\sqrt {a x-b}+a x}+1}{\sqrt {4 b-5}}\right )}{a \sqrt {4 b-5}} \]
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Rubi [A] time = 0.71, antiderivative size = 442, normalized size of antiderivative = 1.78, number of steps used = 25, number of rules used = 14, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6742, 634, 618, 206, 628, 612, 621, 989, 982, 1019, 1076, 1, 1025, 1024} \begin {gather*} \frac {\sqrt {\sqrt {a x-b}+a x} \left (2 \sqrt {a x-b}+1\right )}{2 a}-\frac {2 \sqrt {a x-b}}{a}-\frac {2 \sqrt {\sqrt {a x-b}+a x}}{a}+\frac {\log \left (-\sqrt {a x-b}-a x+1\right )}{a}+\frac {2 \tanh ^{-1}\left (\sqrt {\sqrt {a x-b}+a x}\right )}{a}+\frac {2 (3-2 b) \tanh ^{-1}\left (\frac {2 \sqrt {a x-b}+1}{\sqrt {5-4 b}}\right )}{a \sqrt {5-4 b}}-\frac {(1-4 b) \tanh ^{-1}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )}{4 a}+\frac {2 (1-b) \tanh ^{-1}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )}{a}+\frac {\tanh ^{-1}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )}{a}-\frac {4 (1-b) \tanh ^{-1}\left (\frac {2 \sqrt {a x-b}+1}{\sqrt {5-4 b} \sqrt {\sqrt {a x-b}+a x}}\right )}{a \sqrt {5-4 b}}-\frac {2 \tanh ^{-1}\left (\frac {2 \sqrt {a x-b}+1}{\sqrt {5-4 b} \sqrt {\sqrt {a x-b}+a x}}\right )}{a \sqrt {5-4 b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1
Rule 206
Rule 612
Rule 618
Rule 621
Rule 628
Rule 634
Rule 982
Rule 989
Rule 1019
Rule 1024
Rule 1025
Rule 1076
Rule 6742
Rubi steps
\begin {align*} \int \frac {\sqrt {-b+a x}}{1+\sqrt {a x+\sqrt {-b+a x}}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {x^2}{1+\sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{a}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (-1+\frac {-1+b+x}{-1+b+x+x^2}+\sqrt {b+x+x^2}+\frac {(1-b) \sqrt {b+x+x^2}}{-1+b+x+x^2}-\frac {x \sqrt {b+x+x^2}}{-1+b+x+x^2}\right ) \, dx,x,\sqrt {-b+a x}\right )}{a}\\ &=-\frac {2 \sqrt {-b+a x}}{a}+\frac {2 \operatorname {Subst}\left (\int \frac {-1+b+x}{-1+b+x+x^2} \, dx,x,\sqrt {-b+a x}\right )}{a}+\frac {2 \operatorname {Subst}\left (\int \sqrt {b+x+x^2} \, dx,x,\sqrt {-b+a x}\right )}{a}-\frac {2 \operatorname {Subst}\left (\int \frac {x \sqrt {b+x+x^2}}{-1+b+x+x^2} \, dx,x,\sqrt {-b+a x}\right )}{a}+\frac {(2 (1-b)) \operatorname {Subst}\left (\int \frac {\sqrt {b+x+x^2}}{-1+b+x+x^2} \, dx,x,\sqrt {-b+a x}\right )}{a}\\ &=-\frac {2 \sqrt {-b+a x}}{a}-\frac {2 \sqrt {a x+\sqrt {-b+a x}}}{a}+\frac {\sqrt {a x+\sqrt {-b+a x}} \left (1+2 \sqrt {-b+a x}\right )}{2 a}+\frac {\operatorname {Subst}\left (\int \frac {1+2 x}{-1+b+x+x^2} \, dx,x,\sqrt {-b+a x}\right )}{a}+\frac {2 \operatorname {Subst}\left (\int \frac {\frac {1}{2} (-1+b)-\frac {x}{2}+\frac {x^2}{2}}{\left (-1+b+x+x^2\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{a}-\frac {(1-4 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{4 a}-\frac {(3-2 b) \operatorname {Subst}\left (\int \frac {1}{-1+b+x+x^2} \, dx,x,\sqrt {-b+a x}\right )}{a}+\frac {(2 (1-b)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{a}+\frac {(2 (1-b)) \operatorname {Subst}\left (\int \frac {1}{\left (-1+b+x+x^2\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{a}\\ &=-\frac {2 \sqrt {-b+a x}}{a}-\frac {2 \sqrt {a x+\sqrt {-b+a x}}}{a}+\frac {\sqrt {a x+\sqrt {-b+a x}} \left (1+2 \sqrt {-b+a x}\right )}{2 a}+\frac {\log \left (1-a x-\sqrt {-b+a x}\right )}{a}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{a}+\frac {2 \operatorname {Subst}\left (\int \frac {\frac {1-b}{2}+\frac {1}{2} (-1+b)-x}{\left (-1+b+x+x^2\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{a}-\frac {(1-4 b) \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1+2 \sqrt {-b+a x}}{\sqrt {a x+\sqrt {-b+a x}}}\right )}{2 a}+\frac {(2 (3-2 b)) \operatorname {Subst}\left (\int \frac {1}{5-4 b-x^2} \, dx,x,1+2 \sqrt {-b+a x}\right )}{a}+\frac {(4 (1-b)) \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1+2 \sqrt {-b+a x}}{\sqrt {a x+\sqrt {-b+a x}}}\right )}{a}-\frac {(4 (1-b)) \operatorname {Subst}\left (\int \frac {1}{1-4 (-1+b)-x^2} \, dx,x,\frac {1+2 \sqrt {-b+a x}}{\sqrt {a x+\sqrt {-b+a x}}}\right )}{a}\\ &=-\frac {2 \sqrt {-b+a x}}{a}-\frac {2 \sqrt {a x+\sqrt {-b+a x}}}{a}+\frac {\sqrt {a x+\sqrt {-b+a x}} \left (1+2 \sqrt {-b+a x}\right )}{2 a}+\frac {2 (3-2 b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{\sqrt {5-4 b}}\right )}{a \sqrt {5-4 b}}-\frac {(1-4 b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{4 a}+\frac {2 (1-b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{a}-\frac {4 (1-b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{\sqrt {5-4 b} \sqrt {a x+\sqrt {-b+a x}}}\right )}{a \sqrt {5-4 b}}+\frac {\log \left (1-a x-\sqrt {-b+a x}\right )}{a}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1+2 \sqrt {-b+a x}}{\sqrt {a x+\sqrt {-b+a x}}}\right )}{a}-\frac {2 \operatorname {Subst}\left (\int \frac {x}{\left (-1+b+x+x^2\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{a}\\ &=-\frac {2 \sqrt {-b+a x}}{a}-\frac {2 \sqrt {a x+\sqrt {-b+a x}}}{a}+\frac {\sqrt {a x+\sqrt {-b+a x}} \left (1+2 \sqrt {-b+a x}\right )}{2 a}+\frac {2 (3-2 b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{\sqrt {5-4 b}}\right )}{a \sqrt {5-4 b}}+\frac {\tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{a}-\frac {(1-4 b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{4 a}+\frac {2 (1-b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{a}-\frac {4 (1-b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{\sqrt {5-4 b} \sqrt {a x+\sqrt {-b+a x}}}\right )}{a \sqrt {5-4 b}}+\frac {\log \left (1-a x-\sqrt {-b+a x}\right )}{a}+\frac {\operatorname {Subst}\left (\int \frac {1}{\left (-1+b+x+x^2\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{a}-\frac {\operatorname {Subst}\left (\int \frac {1+2 x}{\left (-1+b+x+x^2\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{a}\\ &=-\frac {2 \sqrt {-b+a x}}{a}-\frac {2 \sqrt {a x+\sqrt {-b+a x}}}{a}+\frac {\sqrt {a x+\sqrt {-b+a x}} \left (1+2 \sqrt {-b+a x}\right )}{2 a}+\frac {2 (3-2 b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{\sqrt {5-4 b}}\right )}{a \sqrt {5-4 b}}+\frac {\tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{a}-\frac {(1-4 b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{4 a}+\frac {2 (1-b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{a}-\frac {4 (1-b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{\sqrt {5-4 b} \sqrt {a x+\sqrt {-b+a x}}}\right )}{a \sqrt {5-4 b}}+\frac {\log \left (1-a x-\sqrt {-b+a x}\right )}{a}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {a x+\sqrt {-b+a x}}\right )}{a}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{1-4 (-1+b)-x^2} \, dx,x,\frac {1+2 \sqrt {-b+a x}}{\sqrt {a x+\sqrt {-b+a x}}}\right )}{a}\\ &=-\frac {2 \sqrt {-b+a x}}{a}-\frac {2 \sqrt {a x+\sqrt {-b+a x}}}{a}+\frac {\sqrt {a x+\sqrt {-b+a x}} \left (1+2 \sqrt {-b+a x}\right )}{2 a}+\frac {2 \tanh ^{-1}\left (\sqrt {a x+\sqrt {-b+a x}}\right )}{a}+\frac {2 (3-2 b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{\sqrt {5-4 b}}\right )}{a \sqrt {5-4 b}}+\frac {\tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{a}-\frac {(1-4 b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{4 a}+\frac {2 (1-b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{a}-\frac {2 \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{\sqrt {5-4 b} \sqrt {a x+\sqrt {-b+a x}}}\right )}{a \sqrt {5-4 b}}-\frac {4 (1-b) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{\sqrt {5-4 b} \sqrt {a x+\sqrt {-b+a x}}}\right )}{a \sqrt {5-4 b}}+\frac {\log \left (1-a x-\sqrt {-b+a x}\right )}{a}\\ \end {align*}
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Mathematica [B] time = 0.79, size = 503, normalized size = 2.03 \begin {gather*} \frac {-8 \sqrt {5-4 b} \sqrt {a x-b}-6 \sqrt {5-4 b} \sqrt {\sqrt {a x-b}+a x}+4 \sqrt {5-4 b} \sqrt {a x-b} \sqrt {\sqrt {a x-b}+a x}+4 \sqrt {5-4 b} \log \left (-\sqrt {a x-b}-a x+1\right )-8 (2 b-3) \tanh ^{-1}\left (\frac {2 \sqrt {a x-b}+1}{\sqrt {5-4 b}}\right )-\sqrt {5-4 b} (4 b-11) \tanh ^{-1}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )-4 \left (2 b+\sqrt {5-4 b}-3\right ) \tanh ^{-1}\left (\frac {-2 \sqrt {5-4 b} \sqrt {a x-b}-4 b-\sqrt {5-4 b}+1}{4 \sqrt {\sqrt {a x-b}+a x}}\right )+8 b \tanh ^{-1}\left (\frac {2 \sqrt {5-4 b} \sqrt {a x-b}-4 b+\sqrt {5-4 b}+1}{4 \sqrt {\sqrt {a x-b}+a x}}\right )-4 \sqrt {5-4 b} \tanh ^{-1}\left (\frac {2 \sqrt {5-4 b} \sqrt {a x-b}-4 b+\sqrt {5-4 b}+1}{4 \sqrt {\sqrt {a x-b}+a x}}\right )-12 \tanh ^{-1}\left (\frac {2 \sqrt {5-4 b} \sqrt {a x-b}-4 b+\sqrt {5-4 b}+1}{4 \sqrt {\sqrt {a x-b}+a x}}\right )}{4 a \sqrt {5-4 b}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.56, size = 232, normalized size = 0.94 \begin {gather*} -\frac {2 \sqrt {-b+a x}}{a}+\frac {\sqrt {a x+\sqrt {-b+a x}} \left (-3+2 \sqrt {-b+a x}\right )}{2 a}-\frac {4 (-3+2 b) \tan ^{-1}\left (\frac {1-2 \sqrt {-b+a x}+2 \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-5+4 b}}\right )}{a \sqrt {-5+4 b}}+\frac {(-19+4 b) \log \left (a \left (-1-2 \sqrt {-b+a x}\right )+2 a \sqrt {a x+\sqrt {-b+a x}}\right )}{4 a}+\frac {2 \log \left (1-2 b-2 (-b+a x)+\sqrt {a x+\sqrt {-b+a x}} \left (-1+2 \sqrt {-b+a x}\right )\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.07, size = 395, normalized size = 1.59 \begin {gather*} \frac {1}{2} \, \sqrt {a x + \sqrt {a x - b}} {\left (\frac {2 \, \sqrt {a x - b}}{a} - \frac {3}{a}\right )} + \frac {{\left (4 \, b - 11\right )} \log \left ({\left | -2 \, \sqrt {a x - b} + 2 \, \sqrt {a x + \sqrt {a x - b}} - 1 \right |}\right )}{4 \, a} + \frac {2 \, {\left (2 \, b - 3\right )} \arctan \left (-\frac {2 \, \sqrt {a x - b} - 2 \, \sqrt {a x + \sqrt {a x - b}} + 3}{\sqrt {4 \, b - 5}}\right )}{a \sqrt {4 \, b - 5}} - \frac {2 \, {\left (2 \, b - 3\right )} \arctan \left (-\frac {2 \, \sqrt {a x - b} - 2 \, \sqrt {a x + \sqrt {a x - b}} - 1}{\sqrt {4 \, b - 5}}\right )}{a \sqrt {4 \, b - 5}} + \frac {2 \, {\left (2 \, b - 3\right )} \arctan \left (\frac {2 \, \sqrt {a x - b} + 1}{\sqrt {4 \, b - 5}}\right )}{a \sqrt {4 \, b - 5}} + \frac {\log \left (a x + \sqrt {a x - b} - 1\right )}{a} - \frac {\log \left ({\left (\sqrt {a x - b} - \sqrt {a x + \sqrt {a x - b}}\right )}^{2} + b + 3 \, \sqrt {a x - b} - 3 \, \sqrt {a x + \sqrt {a x - b}} + 1\right )}{a} + \frac {\log \left ({\left (\sqrt {a x - b} - \sqrt {a x + \sqrt {a x - b}}\right )}^{2} + b - \sqrt {a x - b} + \sqrt {a x + \sqrt {a x - b}} - 1\right )}{a} - \frac {2 \, \sqrt {a x - b}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.72, size = 1551, normalized size = 6.25
method | result | size |
derivativedivides | \(\frac {-\sqrt {\left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )^{2}-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )+1}+\frac {3 \ln \left (\sqrt {a x -b}+\frac {1}{2}+\sqrt {\left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )^{2}-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )+1}\right )}{2}-\sqrt {\left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )^{2}+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )+1}+\frac {3 \ln \left (\sqrt {a x -b}+\frac {1}{2}+\sqrt {\left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )^{2}+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )+1}\right )}{2}-2 \sqrt {a x -b}-\frac {\ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{4}+\ln \left (a x +\sqrt {a x -b}-1\right )+\frac {2 b \arctanh \left (\frac {2+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )}{2 \sqrt {\left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )^{2}+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )+1}}\right )}{\sqrt {5-4 b}}-\frac {2 b \arctanh \left (\frac {2-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )}{2 \sqrt {\left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )^{2}-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )+1}}\right )}{\sqrt {5-4 b}}-\frac {\sqrt {5-4 b}\, \ln \left (\sqrt {a x -b}+\frac {1}{2}+\sqrt {\left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )^{2}+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )+1}\right )}{2}-\frac {6 \arctan \left (\frac {2 \sqrt {a x -b}+1}{\sqrt {-5+4 b}}\right )}{\sqrt {-5+4 b}}+\arctanh \left (\frac {2-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )}{2 \sqrt {\left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )^{2}-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )+1}}\right )+\arctanh \left (\frac {2+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )}{2 \sqrt {\left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )^{2}+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )+1}}\right )-\frac {2 b \sqrt {\left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )^{2}+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )+1}}{\sqrt {5-4 b}}+\frac {2 b \sqrt {\left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )^{2}-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )+1}}{\sqrt {5-4 b}}+\frac {4 \arctan \left (\frac {2 \sqrt {a x -b}+1}{\sqrt {-5+4 b}}\right ) b}{\sqrt {-5+4 b}}+\frac {\sqrt {5-4 b}\, \ln \left (\sqrt {a x -b}+\frac {1}{2}+\sqrt {\left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )^{2}-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )+1}\right )}{2}+\ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right ) b -\frac {3 \sqrt {\left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )^{2}-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )+1}}{\sqrt {5-4 b}}-b \ln \left (\sqrt {a x -b}+\frac {1}{2}+\sqrt {\left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )^{2}-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )+1}\right )+\frac {\left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{2}+\frac {3 \sqrt {\left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )^{2}+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )+1}}{\sqrt {5-4 b}}-b \ln \left (\sqrt {a x -b}+\frac {1}{2}+\sqrt {\left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )^{2}+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )+1}\right )+\frac {3 \arctanh \left (\frac {2-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )}{2 \sqrt {\left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )^{2}-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )+1}}\right )}{\sqrt {5-4 b}}-\frac {3 \arctanh \left (\frac {2+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )}{2 \sqrt {\left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )^{2}+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )+1}}\right )}{\sqrt {5-4 b}}}{a}\) | \(1551\) |
default | \(\frac {-\sqrt {\left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )^{2}-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )+1}+\frac {3 \ln \left (\sqrt {a x -b}+\frac {1}{2}+\sqrt {\left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )^{2}-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )+1}\right )}{2}-\sqrt {\left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )^{2}+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )+1}+\frac {3 \ln \left (\sqrt {a x -b}+\frac {1}{2}+\sqrt {\left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )^{2}+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )+1}\right )}{2}-2 \sqrt {a x -b}-\frac {\ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{4}+\ln \left (a x +\sqrt {a x -b}-1\right )+\frac {2 b \arctanh \left (\frac {2+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )}{2 \sqrt {\left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )^{2}+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )+1}}\right )}{\sqrt {5-4 b}}-\frac {2 b \arctanh \left (\frac {2-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )}{2 \sqrt {\left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )^{2}-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )+1}}\right )}{\sqrt {5-4 b}}-\frac {\sqrt {5-4 b}\, \ln \left (\sqrt {a x -b}+\frac {1}{2}+\sqrt {\left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )^{2}+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )+1}\right )}{2}-\frac {6 \arctan \left (\frac {2 \sqrt {a x -b}+1}{\sqrt {-5+4 b}}\right )}{\sqrt {-5+4 b}}+\arctanh \left (\frac {2-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )}{2 \sqrt {\left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )^{2}-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )+1}}\right )+\arctanh \left (\frac {2+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )}{2 \sqrt {\left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )^{2}+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )+1}}\right )-\frac {2 b \sqrt {\left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )^{2}+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )+1}}{\sqrt {5-4 b}}+\frac {2 b \sqrt {\left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )^{2}-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )+1}}{\sqrt {5-4 b}}+\frac {4 \arctan \left (\frac {2 \sqrt {a x -b}+1}{\sqrt {-5+4 b}}\right ) b}{\sqrt {-5+4 b}}+\frac {\sqrt {5-4 b}\, \ln \left (\sqrt {a x -b}+\frac {1}{2}+\sqrt {\left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )^{2}-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )+1}\right )}{2}+\ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right ) b -\frac {3 \sqrt {\left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )^{2}-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )+1}}{\sqrt {5-4 b}}-b \ln \left (\sqrt {a x -b}+\frac {1}{2}+\sqrt {\left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )^{2}-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )+1}\right )+\frac {\left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{2}+\frac {3 \sqrt {\left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )^{2}+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )+1}}{\sqrt {5-4 b}}-b \ln \left (\sqrt {a x -b}+\frac {1}{2}+\sqrt {\left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )^{2}+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )+1}\right )+\frac {3 \arctanh \left (\frac {2-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )}{2 \sqrt {\left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )^{2}-\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}+\frac {\sqrt {5-4 b}}{2}\right )+1}}\right )}{\sqrt {5-4 b}}-\frac {3 \arctanh \left (\frac {2+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )}{2 \sqrt {\left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )^{2}+\sqrt {5-4 b}\, \left (\sqrt {a x -b}+\frac {1}{2}-\frac {\sqrt {5-4 b}}{2}\right )+1}}\right )}{\sqrt {5-4 b}}}{a}\) | \(1551\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x - b}}{\sqrt {a x + \sqrt {a x - b}} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {a\,x-b}}{\sqrt {a\,x+\sqrt {a\,x-b}}+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x - b}}{\sqrt {a x + \sqrt {a x - b}} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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